A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure

We propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure. These properties allow for accurate computations of stationary states and long-time asymptotics demonstrated by suitably chosen test cases in which these features of the scheme are essential. The proposed scheme is able to cope with non-smooth stationary states, different time scales including metastability, as well as concentrations and self-similar behavior induced by singular nonlocal kernels. We use the scheme to explore properties of these equations beyond their present theoretical knowledge

Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential

We present and analyze finite difference numerical schemes for the Allen Cahn/Cahn-Hilliard equation with a logarithmic Flory Huggins energy potential. Both the first order and second order accurate temporal algorithms are considered. In the first order scheme, we treat the nonlinear logarithmic terms and the surface diffusion term implicitly, and update the linear expansive term and the mobility explicitly. We provide a theoretical justification that, this numerical algorithm has a unique solution such that the positivity is always preserved for the logarithmic arguments. In particular, our analysis reveals a subtle fact: the singular nature of the logarithmic term around the values of $-1$ and 1 prevents the numerical solution reaching these singular values, so that the numerical scheme is always well-defined as long as the numerical solution stays similarly bounded at the previous time step. Furthermore, an unconditional energy stability of the numerical scheme is derived, without any restriction for the time step size. The unique solvability and the positivity-preserving property for the second order scheme are proved using similar ideas, in which the singular nature of the logarithmic term plays an essential role. For both the first and second order accurate schemes, we are able to derive an optimal rate convergence analysis, which gives the full order error estimate. The case with a non-constant mobility is analyzed as well. We also describe a practical and efficient multigrid solver for the proposed numerical schemes, and present some numerical results, which demonstrate the robustness of the numerical schemes

A Fast Semi-implicit Method for Anisotropic Diffusion

Simple finite differencing of the anisotropic diffusion equation, where diffusion is only along a given direction, does not ensure that the numerically calculated heat fluxes are in the correct direction. This can lead to negative temperatures for the anisotropic thermal diffusion equation. In a previous paper we proposed a monotonicity-preserving explicit method which uses limiters (analogous to those used in the solution of hyperbolic equations) to interpolate the temperature gradients at cell faces. However, being explicit, this method was limited by a restrictive Courant-Friedrichs-Lewy (CFL) stability timestep. Here we propose a fast, conservative, directionally-split, semi-implicit method which is second order accurate in space, is stable for large timesteps, and is easy to implement in parallel. Although not strictly monotonicity-preserving, our method gives only small amplitude temperature oscillations at large temperature gradients, and the oscillations are damped in time. With numerical experiments we show that our semi-implicit method can achieve large speed-ups compared to the explicit method, without seriously violating the monotonicity constraint. This method can also be applied to isotropic diffusion, both on regular and distorted meshes.Comment: accepted in the Journal of Computational Physics; 13 pages, 7 figures; updated to the accepted versio

Spatially partitioned embedded Runge-Kutta Methods

We study spatially partitioned embedded Runge–Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in non-embedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to non-physical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical experiments are provided to support the theory

Flux form Semi-Lagrangian methods for parabolic problems

A semi-Lagrangian method for parabolic problems is proposed, that extends previous work by the authors to achieve a fully conservative, flux-form discretization of linear and nonlinear diffusion equations. A basic consistency and convergence analysis are proposed. Numerical examples validate the proposed method and display its potential for consistent semi-Lagrangian discretization of advection--diffusion and nonlinear parabolic problems

Residual equilibrium schemes for time dependent partial differential equations

Many applications involve partial differential equations which admits nontrivial steady state solutions. The design of schemes which are able to describe correctly these equilibrium states may be challenging for numerical methods, in particular for high order ones. In this paper, inspired by micro-macro decomposition methods for kinetic equations, we present a class of schemes which are capable to preserve the steady state solution and achieve high order accuracy for a class of time dependent partial differential equations including nonlinear diffusion equations and kinetic equations. Extension to systems of conservation laws with source terms are also discussed.Comment: 23 pages, 12 figure